416 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
Aggregation of Oscillating Subsystems
R.Joel Rahn
Faculté des sciences de l’ administration
Université Laval
Ste-Foy Québec GiK 7P4 CANADA
ABSTRACT
In previous work, the analysis of the effects of aggregating simple dynamic
systems has been studied by applying methods developed for thermodynamic
systems in order ta take account of stochastic effects. This approach is
based on the Master Equation for the probability density of the contents of
@ vector of system levels. The goal of these studies is to determine the
dynamic characteristics of systems composed of a population of sub-systems
with the same dynamic structure while accounting for novel behavior that is
introduced by the process of aggregating the sub-systems into the larger
system.
In this paper, the Master Equation analysis is applied to four versions
of a Commodity Cycle model to determine the nature of modes of
behavior that arise from the process of aggregating a population of
entities whose dynamic structure is derived from the oscillatory
structure of the commodity cycle model. The approach used here is novel
in two respects. it contrasts with the more recently developed
analysis of chaotic systems in which non-linear, aggregate cr luaped —
Parameter models generate behavior that is unpredictable while not
being stochastic. In those models, no attempt is made to explain the
large-scale or aggregate chaotic behavior in terms of the sub-systems.
Compared to previous work in the same vein, this paper addresses itself
to a slightly larger model as part of a natural progression in the
analysis of ever-more complex systems by Master Equaticn methods.
INTRODUCTION
One of the fundamental problems underlying the analysis of complex
systems is the relationship between the structure and behavicr of the
aggregate system and the structure and behavicr of the sub-systems
which comprise the larger system. In previous papers on this theme
(Rahn 1985, Rahn 1983), the author has proposed an approach based on an
interpretation of the system dynamic equations as representing the
lowest order term in an expansion of the Master Equation for the joint
probability distribution of the levels comprising the system state. The
result of the expansion is a consistent treatment of the equations for
the most-probable evolution path of the system and for the mean and
variance of fluctuations about this path (van Kampen 1961, Tomita et al
1974, Portnow et al 1976).
The present paper applies the same methods to four small models of the
behavior of commodity cycles (Meadows 1970, Gocdman 1974). These
models all include an oscillatory structure but differ in their
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA-417
treatment of non-linear structure, number of levels and nature of other
loops (minor and major). The analysis of these models completes
another step in the research program proposed in (Rahn 1985).
As in the previously mentioned analyses, some general characteristics
of the method are demonstrated. For example, to lowest order in the
expansion scheme used, the fluctuations about the most probable path
behave similarly to the macroscopic system. That is, in linear systeas
the eigenvalues of the fluctuation equations are the same as for the
most-probable path; in non-linear systems, the variance of the
fluctuations grows during periods of expansion or local instability and
stabilizes when the macrocsopic system reaches equilibrium. More
detailed evaluation of the characteristics of the fluctuations is
reserved for the discussion of each case.
MODEL I
The first model is composed of two levels representing mature and juvenile
stocks of animals linked by a maturation delay. The birth rate of juveniles
is an increasing function of the market price for mature animals and is
proportional to the number of mature animals. The. consumption rate of
mature animals is a decreasing function of the market price and proportional
to the (constant) population. The equations of the model in differential
form are:
Y = -mY + FBRN*®FBRM(P) «A JUVENILES 7
A = aY — POP*PCC(P) ADULTS
PIA) = por BA MARKET PRICE Ww
FBRM{P) = byth,P FRACTIONAL BIRTH RATE MULTIPLIER
PCC(P) = fy ~ c,P PER CAPITA CONSUMPTION
In this form, non-linear table functions for P, FBRM and PCC are linearized
about the current value of the level vector. The factor @ in the dynamic
equations is the reciprocal of the maturation delay time. Substituting
Single symbols for complicated combinations of constants, we can write the
dynamic equations more simply as:
Y= ay 47,8 - yaa”
A= MY ~aA-a
ta)
9
To interpret these equations as defining a stechastic birth and
death process, we define a set of elementary tragsitions or changes in the
number of Y and A elements and the associated probabilities of such
transitions. In this model, it is natural te specify that the transition
418 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY, CHINA
that removes one unit of Y (by maturation) simultaneously adds one unit to
43 i.e. the transitions are not independent of each other. With this
specification, we have the transition table shown below where x is the
vector of levels {Y,A) and 6x is the transition vector.
Transition Probability w(x»dx)
6Y=+1,6R= 0 yA ~ wae
6Y= O,6A = +1 aAta
6Y=-1,6h = +1 mY
From these specifications, we derive the macrosopic rate of change for the
most-probable path (maximum of the distribution of the state vector, y):
c= EF 6Yalysdx) nay +, — ygA®
of SY 43)
=f GAwly,dx) mY — aA — &
SA BY
We see that the macroscopic equation reproduces the original system
equations as expected. From equation 3, we can derive the system matrix
for the mean of the fluctuations, ps
8c} ty) -a y, - ®y,8
=i. [ i 2 4)
ry -a
The system matrix for the variance-covariance matrix of the fluctuations is
found from the equation:
do Ko + (Koo? + co)
dt
where D is the @atrix of second moments of the transition probabilities and
is a function of y» the state vector of the most-probable path. The
characteristics of the mean and covariances are summarized by the
eigenvalues of the corresponding system matrices. For the mean, we have
the characteristic equation:
Peta, emrimla,—ly,-2y,A) 20 ()
from which we see that there are oscillations iff
2 _ 2,
(ay +m)" <ama,—Cy,-2ygAY) or a, -m)" <amtBy,A-y, ) a7)
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 419
and there is an unstable root if
wamia, — ty, ~ BygAd) > 0
or equivalently if
-a
i
Ye
y
o< ac tt
The numerator of the right-hand inequality can be re-written in teras of |
the orginal model variables as:
FERN. (by + by-By) > POP.C,.p,
This latter relationship states the reasonable condition that unstable
growth modes can occur if the birth rate is greater than the consumption
rate while the limit on the size of the mature population means that the
instability will be limited by the growth of the mature population level.
For the covariances, the eigenvalue equation is
rw # a. +m + AITCA + BMA + Bay) ~ 4ad,3 = 0
rT
whose solutions are:
f
A= = fay tas d = ~ta testa, 4m) Pend 1?
where dy = y¥,~2y,A
Comparison of this result with equation 7 shows that the condition for
oscillations to occur is less restrictive for the mean than for the
covariances of the fluctuations. We see also that the covariances are
unstable if
< Amd, = 4mly,-2y QA) < amy -y,A)
i.e.) instability may occur only for small values of the mature
population, at which time the system is in a growth phase.
MODEL IT
The second model is a siaplified version of a commodity cycle model in
{Goodman 1974); the main simplification consists in suppressing the use of
an expected market price in favor of the market price itself. This
modification eliminates a delay and thus a minor negative loop. The
dynamic equations are:
PC = (DPC - PC)/CAD PRODUCTION CAPACITY
DI = PR - CR DISTRIBUTOR’S INVENTORY
DPC = ds tt: ayP DESIRED PRODUCTION CAPACITY (8)
PR = PC PRODUCTION RATE
CR = POP. icy - cP) CONSUMPTION RATE
P= py ~ p,DI MARKET PRICE
420 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
After substitution we get the system equations for PC, DI:
PC = ido td, (Pg-p DI) -PC) /CAD
I= PC - POP (c-c, (pgp, DEI)
oP)
In this aodel the transitions of PC and DI are mutually independent. me -
other feature of interest is the fact that the single term in the rate of
change of PC can be positive or negative so that allowance must be made for
this possibility in the specification of the transition probabilities. It
will be seen that this feature has no effect on the characteristics of the
dynamics of the mean and covariance of the fluctuations. .
Transition Probability w{x»dx)
SPC = + 1,6D1 = 0 ptPCCDPC) (dots, (pgp, DI)-PC)/CAD
6PC = - 1,;6D1 = 6 BIPEDDPC) (PC (dtd tn -p DI) /CAD
6PC = 0, SDI = +1 PC
6PC = 0, SDI = -1 POP. (cq-, (By-p DED)
The vector of the macroscopic rate of change reproduces the linearized
system equations since the sum of the probabilities:
p(PC<DPC) + p(PO>DPC) = 1.
The system matrix for the sean-of the fluctuations is:
Ke Cr a, p, /CAD ]
1 “POPC Py
From this matrix we can determine that the fluctuations are always stable
and if there are oscillations, they are at the same frequency as the
macroscopic system.
For the variance-covariances of the fluctuations, the eigenvalue equation
is a cubic:
_OSPOPC yp +1 /C8D) [Ot2/CAD) O42.POPC,p, )+4d p /CAD]=0 410)
We note that this equation factors into a real, negative root and a
quadratic term as in Model I and as in the Workforce-Inventory model in
{Rahn 1983) in spite of differences in both the dynamic and the
stochastic models. The robustness of this feature is due to the linearity
of the underlying models. By substituting in equation 10: t,
A/CAR
POPC, Py
aR, /CAD
a
b
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 421
we can re-write it as
(Atatb £(A+2a) (.+2b)44D] = 0 qui)
whose solutions are:
4 = — lath)
Ag = cfatb) + Catto? -acabenya!/? = —asby © c¢aby® - apr?
The mean and covariance components decay exponentially with or without
oscillations under the same conditions as the macroscopic, linear model.
This result was first derived for the tase of a linear Workforce-Inventory
gcedel in (Rahn 1983).
MODEL IIIT
The third model is a simplification of the generic commodity cycle model
presented in (Meadows 1970). The main simplification consists in
eliminating the expected consumption rate function (a delay) in favor of
the consumption rate as a function of market price. The main difference
with respect to the previous model is the inclusion of the expected market
price as an explicit level so that the mcde] has three levels. The dynamic
equations in differential form ares
PC = (DPC - PC) /CAD - PC/ALPE PRODUCTION CAPACITY
DI = PR - CR DISTRIBUTOR INVENTORY
EP = (P - EP)/EPAD EXPECTED HARKET PRICE
P= PIDL) = pop DE MARKET PRICE
DPC = dotd EP DESIRED PRODUCTION CAPACITY (12)
PR = PC PRODUCTION RATE
CR = POPsPCLIP) CONSUMPTION RATE
PCC = eget yP PER CAPITA CONSUMPTION RATE
After substitution of the linearized fores of the table functions. in
equation 12 we get the system equations:
PC = (dgtd ,EP-PC)/CAD — PC/ALPC
DI = PC ~ POPEe (poop DID)
EP = ((po7p,DI) ~ EP) /EPAD
The transitions are again considered to be mutually independent and we
propose the following stochastic model:
422 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
Transition Probability wix,dx)
SPC = + 1,6DI = 0,5EP= 0 PAPC<DPC) (dtd EP-PC) /CAD
OPC = - 1,5DI = 0;6EP= 0 p(PC>DPC) (PC-d-d EP) /CAD+PC/ALPC,
6PC = 0, SDI = +1,6EP= 0 PC
SPC = 0, SDI = -1,S5EP= 0 POP. (coc, (py-p DID)
SPT = 0, SDI = OsSEP= 1 pCEP<P) (pp DI-EP) /EPAD
SPC = 05 GDI = Oy SEP=-1 PIEP>P) (EP-pytp ,DI) /EPAD
The macroscopic rate of change, cy fys reproduces the linearized system
equations for the same reason as in Model II. The system matrix for the
mean of the fluctuations is:
1/CAD — 1/ALPC ° ,/CAD
1
Ke °
ag
° -1/ECAD
The eigenvalue equation for the mean fluctuation, 4, is
{\ + 2/CAD + 1/ALPODID + POPC 1p, )ir + 1/EPAD) - D= 0
The eigenvaiue equation for the variance-covariances is a very complicated
Sixth order polynomial in factors of (A+...) whose constant term is -an®,
Fer both the mean and variances, the constant term acts to perturb the
negative eigenvalue of smallest absolute value in the direction of less
stability, towards smaller absolute values or even to destabilize it. The
constant term, by shifting the eigenvalue function vertically downward, may
Cause some real rpots to coalesce in complex conjugate pairs which are
stable.
MODEL IV
The final model is another simplification of the generic commodity cycle
model in (Meadows 1970). In this version, the expected market price is
replaced by the market price and the expected consumption rate is included
in order to have an explicit non-linearity in the dynamic equations:
Pc = (BPC-PC)/CAD -PC/ALPC PRODUCTION CAPACITY
DI = PC - POP*PCC(P) DISTRIBUTOR INVENTORY
ECR = (CR-ECRI/ECAD EXPECTED CONSUMPTION RATE
Pe PeCov/DEQV) = PooP ,DIVECR HARKET PRICE
PCC = Cg t, (PoP DI/ECR) PER CAPITA CONSUMPTION RATE
DPC = dtd P DESIRED PRODUCTION CAPACITY
qt
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 423
The system matrix for the @ean of the fluctuations is
“1/CAD-1/0LPC -D, D,DI/ECR
1 *, P bI/ECR
Kae 0 P,/ECAD -(P,DI/ECR + 1)/ECAD
where w« Ke
D, = 4,p,/(ECR.CAD)3 P, = POPC,p, /ECR
The stability of the fluctuations is determined by a complicated cubic
whose constant term is *destabilizing’ in the sense mentioned previously
for Model III; i.e. the real, negative root of smallest absolute value is
displaced to the right by the downward shift imposed by the constant term.
CONCLUSIONS:
In all of these commodity cycle models except for Model 11, there is a
possibility for the mean and/or the variance-covariances of the
fluctuations to become unstable. Such a development should make itself
evident as a modification of the macroscopic behavior mode unless other
factors such as non-linearity disrupt the behavior expected on the basis of
the foregoing analysis. The modest range of models that has been
presented in this series of papers indicates that there are some
situations in which fluctuations may grow exponentially. It remains to
verify these possibilities by simulation.
Models of third order for the mean of the fluctuations about the
most-probable path seem to be at the limit of intuitive analysis especially
as regards the evolution of the variance-covariances which give a sixth
order polynomial for the eigenvalue equation. Further effort to analyze
more realistic models will require more powerful tools.
REFERENCES
Goodman, M.R. (1974), Study Notes in System Dynamics, Wright-Allen
Press, Cambridge NA
Meadows, D.L.M. (1970), Dynamics of Commodity Production Oycles, MIT
Press, Cambridge MA -
Portnow, J., K. Kitahara (1976), “Stability and Fluctuations in
Chemical Systems", J.Stat Physics, vol. 14, p. 501
Rahn, R.J. (1985), “Aggregation in System Dynamics", Systes Dynamics
Review, vol.1, no.i
Rahn, R.J. (1983), "Some Dynamic Effects of the Aggregation of Generic
Model Systems - The Master Equation Approach" in Proceedings of the
1983 International System Dynamics Conference, Chestmut Hill MA, July
Tomita, K.» T. Ghta, H. Tomita (1974), “Irreversible Circulation ars
Orbital Revolution", Prog. Theor.Physics, vol 52, p. 1744
van Kampen, N.G. (1961), "A Power Series Expansion of the Master
Equation", Cas Physics, vol. 39, ¢.352
